Three methods to generalize Pawlak approximations via simply open concepts with economic applications

The generalizations of rough sets based on topological structures have become the hot topic of rough set theory. In the present article, based on the notion of j\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j$$\end{document}-neighborhood space and the related concept of j\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j$$\end{document}-simply open sets, three new methods for generalizing Pawlak rough sets are proposed and their properties are studied. These methods are compared with some of the previous methods. Moreover, we prove that the suggested approaches are more accurate than the other methods and then these approaches may be useful in real-life applications. Finally, an economic application in decision-making is introduced as a simple practical example to demonstrate the ideas proposed. It is providing a comparison between the proposed methods with already existing in the literature.

In (1982), Pawlak proposed the theory of rough sets for handling ambiguity, imprecision, and uncertainty in data analysis. Established on the equivalence relation that generating from the data, he suggested the lower and upper approximations for dealing with the vagueness (ambiguity) are arising from the rough set. A concept that can consist of ambiguous data is categorized in this principle by a couple of two definite sets (namely, its lower and upper approximations). These approximations represent the key notions in the methodology of rough sets, which are well-defined according to equivalence classes. The approximations can be used to extract new details about the concept, for instance, in decision rules (or decision-making), the lower approximation shows the characteristics of objects that are definitely related to the concept, but those created via the upper approximation describe items that are possible to belong to the concept. So, the approximations of undefinable (rough) sets by definable sets are discussed. If the subset of the universe has equal approximations, then it is said to be a crisp (or exact) set. On the other hand, if the lower and upper approximations are not equal, then it is a rough (or inexact) set. Accordingly, the difference between the upper and lower approximation (which represents the boundary region) is the basic tool to identify the accuracy or the vagueness of the set. Consequently, we can identify the exactness or roughness of any subset by detecting the region of the boundary is empty or not, respectively. If the boundary region of a subset is not empty, then this interprets that there is not enough knowledge to describe it. For these reasons, the theory of rough set has numerous achievements in the different fields. However, the basic methodology of Pawlak's philosophy is defining approximations by using a partition generated via an equivalence relation. This relation has constituted an impediment to the field of application because it is only dealing with an information system with whole information. So, to handle complex and difficult applied problems, many authors proposed numerous generalizations (for instance, the reader can see (El-Atik 1997;Neubrunnova 1975;Mashhour et al. 1982;Andrijević and Semi-preopen sets, ibid. 1986;Andrijević 1996;Abd El-Monsef 1980;Abd El-Monsef et al. 1983, 2014aLevine 1963)) to expand the application fields of this theory.
To extend Pawlak rough sets, Yao (996) used an arbitrary relation for constructing lower and upper approximations without extra conditions on the relation. However, the main properties of Pawlak's models did not hold for his approximations. Abd El-Monsef et al. (2014a) proposed a structure to generalize the standard rough set concept. In fact, they presented the notion of j-neighborhood space (briefly, j-NS) constructed by a binary relation. Moreover, they used a general topology generated from the binary relation to construct generalized rough sets. This methodology paved the way for extra topological presentations in the rough set contexts and helped to formalize various applications from daily-life problems. After then, Amer et al. (2017) applied some near open sets in j-NS and thus they succussed in generating new generalized rough set approximations, namely j-near approximations. In 2018, Hosny (2018) extends these approximations to different approximations using the concepts of db-open and^b-open sets.
The core contribution of the present work is to propose three different methods for generalizing Pawlak rough sets and some of their generalizations, to reduce the boundary region and enlarge the accuracy degree which represents the core purpose of rough set philosophy. Therefore, the application fields of the rough set methodology are extended.
Firstly, in Sect. 3, we apply the concept of j-simply open sets in the j-NS and then new types of near open sets are defined. Properties of these sets are studied and their connections with the other types are investigated. Several counterexamples are suggested to illustrate those relationships. Additionally, some vital relationships and results of those sets are demonstrated. By applying the concept of jsimply open sets in the j-NS, new types of near open sets are defined, in Sect. 4. Thus, Sect. 4 introduces three different methods to generalize Pawlak's models established by j-simply open sets. The first method of our approaches is based on these types of j-simply open sets, and hence, we propose generalized rough approximations, namely j-simply approximations as a generalization to Pawlak rough sets. We prove that these approximations satisfy all properties of Pawlak's rough sets. Moreover, we illustrate that this method is a generalization to Pawlak's rough sets and their generalizations. Some results are introduced to demonstrate that our approximations are stronger than the other methods. Accordingly, we introduce a comparison between our method and the other methods.
By  Amer et al., and M. Hosny techniques) are established in the other two methods. Moreover, we will prove that the suggested methods strengthen the concept of the previous methods. Many comparisons between the proposed methods and the other ones are investigated in order to illustrate these facts. Therefore, the proposed methods may be convenient in the rough set environment for removing the ambiguity, and thus, it will be beneficial in decision-making.
Finally, in Sect. 5, an application example in economics is presented to explain the importance of the suggested techniques in decision-making. Besides, the proposed methods are compared with the previous methods. In fact, we use an information system (with decision attributes) collected from a real-life problem. This system depends basically on a reflexive relation which means that Pawlak's rough sets can't apply here. So, we apply our methods and the previous methods. We will show that the suggested methods are stronger and more accurate than the other methods. Therefore, we can say that our methods extend the application field for rough sets from a topological standpoint.

Basic concepts
The present section presents the necessary concepts that were used through the paper.

Pawlak rough set theory and its generalization
Definition 2.1 (Pawlak 1982) Consider the relation R is an equivalence relation on a finite set U called universe. The pair U; R ð Þ is called an approximation space. For any X U, Pawlak proposed the lower and upper approxima- , respectively, where x ½ R denotes to an equivalence class consists of x 2 U.
Note that Pawlak (1982): X is said to be a rough set if R X ð Þ 6 ¼ R X ð Þ. Otherwise, it is definable (or exact).
Proposition 2.1 Pawlak (1982) Suppose that U; R ð Þ be an approximation space, ; represents an empty set and X c represents a complement of X in U. Then, the followings are held: Yao (Yao 1996) have extended Pawlak approximation space by a generalized binary relation. In fact, he suggested rough approximations via the after set of a binary relation as shown in the next definition.
Definition 2.2 (1996) Consider R be any binary relation on U. The lower and upper approximations of A U are given, respectively, by where n x ð Þ represents the after set of x, that is, n x ð Þ ¼ y 2 U : xRy f g¼ xR. The boundary region and accuracy of the approximations are given, respectively, by where R n A ð Þ 6 ¼ 0.

j-Neighborhood space and j-approximations
Abd El-Monsef et al. (2014a) introduced a framework to generalize Pawlak's rough sets concept. In fact, they suggested different approximations based on a topological structure.
Definition 2.3 (Abd El-Monsef et al. 2014a) Suppose that R be an arbitrary binary relation on U. The j-neighborhood of x 2 U, (denoted by N j x ð Þ; 8j 2 r; '; r; '; u; i; hui; hii f g ), is given by Definition 2.4 (Abd El-Monsef et al. 2014a) Suppose that R be an arbitrary binary relation on U and a map n j : U ! P U ð Þ assigns for each x 2 U its N j x ð Þ in the power set P U ð Þ. Then, the triple U; R; n j À Á is said to be j-neighborhood space (briefly, j-NS).
The complement of a j-open set is called j-closed and the family C j of all j-closed sets is given by C j ¼ F UjF c 2 s j È É and the j-interior (resp. j-closure) operator of A is given by Definition 2.6 (Abd El-Monsef et al. 2014a) Let U; R; n j À Á be a j-NS, and A U. Then, the (j-lower and j-upper) approximations, (j-boundary, j-positive and j-negative) regions and j-accuracy of the approximations of A U are given, respectively, by

j-Near approximations operators in the j-NS
Amer et al. (Amer et al. 2017) applied the notion of near open sets in the j-NS and thus they succussed in generating new generations to j-approximations, namely j-near approximations.
The previous sets are called j-near open sets and the complement of j-near open is called j-near closed. The family of all j-near open (resp.j-near closed) sets of U is denoted by Definition 2.8 (Amer et al. 2017) Let U; R; n j À Á be a j-NS, j 2 r; '; r; '; u; i; hui; hii f g , k 2 r Ã ; p; s; c; a; b f gand A U. Then, the j-near (lower and upper approximations), j-near (boundary, positive and negative regions) and j-near accuracy the j-near approximations of A are given, respectively, by Theorem 2.2 (Amer et al. 2017) If U; R; n j À Á is a j-NS. Then, for each k 2 p; s; c; a; b f g , and k 6 ¼ r Ã : Proposition 2.3 (Amer et al. 2017) Let U; R; n j À Á be a j-NS, and A U. Then, the followings are held.
be a j-NS, and A U. Then, the followings are held.
M. Hosny (Hosny 2018) extend the above approximations into different approximations via the concepts of dbopen sets and^b-open sets as shown in the following definitions.
Definition 2.9 (Hosny 2018) If U; R; n j À Á is a j-NS and A U. Then: Definition 2.10 (Hosny 2018) Let U; R; n j À Á be a j-NS and A U. Then, the (db j -lower and db j -upper) approximations, (db j -boundary, db j -positive and db j -negative) regions and db j -accuracy of the approximations of A are given, respectively, by Definition 2.11 (Hosny 2018) Let U; R; n j À Á be a j-NS, and Definition 2.12 (Hosny 2018) Let U; R; n j À Á be a j-NS and A U. Then, the (^b j -lower and^b j -upper) approximations, (^b j -boundary,^b j -positive and^b j -negative) regions and^b j -accuracy of the approximations of A are given, respectively, by Proposition 2.6 (Hosny 2018) Let U; R; n j À Á be a j-NS, and A U. Then, the followings are held.
Corollary 2.2 (Hosny 2018) Let U; R; n j À Á be a j-NS, and A U. Then, the followings are held.
3 On simply open set concepts in the jneighborhood space In the present section, we discuss the notion of j-simply open sets in the j-NS and give some more properties of them. Relationships between the suggested sets and the other types of near open sets are examined with counterexamples.
Definition 3.1 Consider U; R; n j À Á be a j-NS and 8j 2 r; '; r; '; u; i; hui; hii The complement of j-simply open is called a j-simply closed and the family of all j-simply open (resp. j-simply closed) sets of U is symbolized by SM j O U ð Þ (resp. SM j C U ð Þ).
Theorem 3.1 Let U; R; n j À Á be a j-NS. Then, the class SM j O U ð Þ is a topology on U.

Proof
(1) Obviously, U and ; are j-simply open sets.
(2) Let But, from the properties of int j and cl j , From (1), (2) and (3), SM j O U ð Þ is a topology on U.
Theorem 3.2 Let U; R; n j À Á be a j-NS. Then, every a jsimply open set is a j-simply closed set and vice versa.
Proof Let A be a j-simply open set, then int j cl j A ð Þ À Á cl j int j A ð Þ À Á . By taking the complement of the both sides, we get: Corollary 3.2 Let U; R; n j À Á be a j-NS. The following statements are true:  Remark 3.3 Let U; R; n j À Á be a j-NS and A U. Then, the followings are not true in general: So, the different types of j-simply open sets are independent and then the relationships among them differ than the relationships among different types of s j .
To illustrate Remark 3.3, we give the following example.
Example 3.2 Consider Example 3.1, the j-neighborhoods of each element in U are given in Table 1 Similarly, one can give another example to illustrate the others statements of Remark 3.3.
Note that: The classes of j-simply open sets and  Several results are proposed to show that the suggested methods are stronger and accurate than the other methods.

First method to rough set approximations
Definition 4.1 If U; R; n j À Á is a j-NS, then the j-simply (lower and upper) approximations, j-simply (boundary, positive and negative) regions and j-simply accuracy of the j-simply approximations of A U are given, respectively, by The core aim of the next consequences is to show the connections between j-simply approximations and some of the other types.
Theorem 4.1 Let U; R; n j À Á be a j-NS, and A U. Then, We only verify (i), the other items can be made likewise.
Let x 2 R j A ð Þ, then 9G 2 s j such that G A and x 2 G. But, from Corollary 3.2, every j-open set is j-simply open, hence G is j-simply open and G 2 SM j O U ð Þ such that G A and x 2 G. Therefore, x 2 R sm j A ð Þ.
Theorem 4.2 Let U; R; n j À Á be a j-NS. Then, 8A U: Corollary 4.1 Let U; R; n j À Á be a j-NS, and A U. Then, Remark 4.1 The opposite of the previous consequences is not correct in overall as exposed in Example 4.1.
Basic properties of j-simply approximations are given in the following proposition.
Proposition 4.1 If U; R; n j À Á is a j-NS and X; Y U. Then: Proof From Theorem 3.1 and Corollary 3.1, the class SM j O U ð Þ represents a quasi-discrete topology. Then, R sm j X ð Þ and R sm j X ð Þ represent the interior and closure operators of X. Thus, the properties (L1-L11) and (U1-U11) are held.
Remark 4.2 Proposition 4.1 illustrates that the j-simply approximations satisfied all of the characteristics of Pawlak's rough approximations without any restrictions. Therefore, we can say that the proposed methodologies in Definition 4.1 represent the natural generalization to Pawlak's model and some of its generalizations.
The next example explains this remark.

Second method to rough set approximations
Definition 4.2 If U; R; n j À Á is a j-NS. Then, the j-generalized (lower and upper) approximations, (boundary, positive and negative) regions and the j-generalized accuracy of the j-generalized approximations of A U are given, respectively, by Note that: A is called a j-generalized exact set if Obviously, A is j-generalized exact if l 1 j A ð Þ ¼ 1 and BND 1 j A ð Þ ¼ ;. Otherwise, it is j-generalized rough. From Definition 4.2 and Theorem 4.1, it is easy to demonstrate the next consequences, so we omit the proof.
Theorem 4.3 4.3 Let U; R; n j À Á be a j-NS, and A U. Then: Corollary 4.2 Let U; R; n j À Á be a j-NS, and A U. Then, Corollary 4.3 Let U; R; n j À Á be a j-NS, and A U. Then, the following statements are true in general:

Third method to rough set approximations
Definition 4.3 If U; R; n j À Á is a j-NS. Then, the j-generalized (lower and upper) approximations, j-generalized (boundary, positive and negative) regions and the j-generalized accuracy of the approximations of A U are given, respectively, by Obviously, A is j-generalized exact if l 2 j A ð Þ ¼ 1 and Otherwise, it is j-generalized rough. From Definition 4.3 and Theorem 4.1, it is easy to demonstrate the next consequences, so we omit the proof.
Theorem 4.4 4.4 Let U; R; n j À Á be a j-NS, and A U. Then: Corollary 4.4 Let U; R; n j À Á be a j-NS, and A U. Then, Corollary 4.5 Let U; R; n j À Á be a j-NS and A U. Then, the following statements are true in general.
Remark 4.4 The reverse of the previous consequences is not right in overall as exposed in Example 4.3.

Economy application
Economic development in most western countries has been an official policy target since the 1950s. Generally speaking, since the 1970s, growth rates have been slightly slower than throughout the two earlier years. Additionally, after the economic recession in 2008, economic growth has not yet improved in utmost countries. The expectation (still principal) that GDP growth will stay to rise at a regular pace of 2.5% in the next period is now being challenged by a growing number of economists and commentators (Malmaeus and Alfredsson 2017;Mafizur Rahman 2017). Mainstream economists are proposing a new standard yearly growth ratio of 1% or less mainly owing to an inferior predictable rate of industrial progress and thus a lesser productivity growth rate. Zero growth or even disastrous trends are considered outside the norm due to the decrease in oil production, other resource limitations, and negative consequences of the degradation of the environment and climate change. A criterion is an attribute in an information system if the field of a condition attribute is well ordered in decreasing or increasing preference. The set-valued information system depends on each condition's attribute as a criterion. If the objects ordered increase or decrease in choice according to inclusion, then the attribute is an inclusion criterion. As in the example below, national production can be calculated by three techniques. This system depends on a reflexive relation and thus Pawlak's rough sets can't apply here. Accordingly, we apply the suggested methods and the previous methods in this decision system of countries, and then, we compare these decision-making methods.
Note that: The introduced application is constructed from real-life problems in Mafizur Rahman (2017).
Example 5.1 (Mafizur Rahman 2017) Consider U ¼ C 1 ; C 2 ; C 3 ; C 4 ; C 5 f gbe a universe of five countries and A ¼ a 1 ; a 2 ; a 3 f g the set of attributes which measure the national product in these countries, where a 1 stands for a product method, a 2 stands for a spending method and a 3 stands for an income method and decision attribute = { growth, not growth}. Now, suppose that the value sets of the attributes are given by V a 1 ¼ F; T; V f gwhere F, T and V represent {Finished product style, Taxes and Value-added style}.
V a 2 ¼ C; I; G f g where C, I and G represent {Consumption, Investment and Government}.
V a 3 ¼ S; P; R f g where S, P and R represent {Salaries, Profits and Rent}. Table 5 represents an information system with decision attribute d ¼ Growth; Not growth f g . Thus, the relation that represents this system can be given by xR a i y , V a i x ð Þ V a i y ð Þ, for each i 2 1; 2; 3 f g and x; y 2 U.
For the first attribute a 1 , we get: xR a 1 y ¼ C 1 ; C 1 ð Þ; C 1 ; C 2 ð Þ; C 2 ; C 2 ð Þ; C 3 ; C 2 ð Þ; f C 3 ; C 3 ð Þ; C 3 ; C 4 ð Þ; C 4 ; C 4 ð Þ; C 5 ; C 4 ð Þ; C 5 ; C 5 ð Þg. Then, countries set B are Proposition 4.1. Finally, an economic application as a simple example to demonstrate the proposed methods has been introduced. Actually, we illustrated that the suggested approximations are more accurate than the previous methods and then it will be useful in the rough sets' context and extend the application fields of rough set theory. Accordingly, we can say that the suggested structures will provide important tools for the handling of knowledge in information systems. In the future, we will apply the proposed techniques in more real-life applications.
Acknowledgements The authors sincerely thank and appreciate the reviewers for the careful reading and thoughtful comments. The present version of the paper owes much to their precise and valuable remarks. The authors would like to thank all Tanta Topological Seminar colleagues ''under the leadership of Prof. Dr. A. M. Kozae'' for their interest, ongoing encouragement, and lively discussions. They would also like to dedicate this work to the memory of Prof. Dr. M. E. Abd El-Monsef ''God's mercy,'' who passed away in 2014.
Funding Open access funding provided by The Science, Technology & Innovation Funding Authority (STDF) in cooperation with The Egyptian Knowledge Bank (EKB). The authors were not supported by any organization for the work submitted.
Data availability Not data were used to support this study.

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Conflict of interest The authors declare that there is no conflict of interest regarding the publication of this manuscript.
Ethical approval This article does not contain any studies with human participants or animals performed by any of the authors.
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